Let $K=\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_r})$ for square-free integers $a_1,\ldots,a_r$ --a multiquadratic number field. Suppose the generating elements $a_1,\ldots,a_r$ is normalized. For each integer $j\in\{1,\ldots,2^r\}$, there is a unique 2-adic presentation $$j-1=\sum_{i=1}^r j_i2^{i-1}$$ and put $b_j:=a_1^{j_1}\cdots a_r^{j_r}$ and $\gamma_j:=\sqrt{b_j}$. Furthermore, let $g_j\in\mathbb{Z}$ such that $$0\leq v_p(b_j/g_j^2)\leq 1\qquad \text{for each prime }p\qquad (j=1,\cdots,2^r),$$ where $v_p$ is the $p$-adic valuation of $\mathbb{Q}$.
Schmal's Theorem: An integral basis for $K$ is given by $\{\omega_1,\ldots,\omega_{2^r}\}$ where $$\omega_j=\dfrac{1}{2^{\delta_j}g_j}\prod_{i=1}^r(\sqrt{a_i}-a_i)^{\alpha_{ji}}\quad (i\leq j\leq 2^r)$$ $$\delta_1=0, \quad \delta_2=\left\{ \begin{array}{ll} 1 & \text{for }a_1\equiv 1 \pmod 4 \\ 0 & \text{for }a_1\equiv 2,3 \pmod 4 \end{array}\right., \quad \text{and} \quad \delta_j=\sum_{i=1}^r\alpha_{ij}-\beta_j \quad (2<j\leq 2^r)$$ with $$\beta_j=\left\{\begin{array}{ll} 1 & \text{for } (a_1,a_2)\equiv (2,1), (3,1) \pmod 4, j_1=1\\ 1 & \text{for } (a_1,a_2)\equiv (2,3) \pmod 4, j_1=1 \text{ or } j_2=1\\ 0 & \text{else}. \end{array}\right.$$
Using Petho's paper "On the indices of multiquadratic number field", I understood the proof for the case when $gcd(a_1,\ldots,a_r)=1$. Unfortunately, the general case was not discussed by Petho, while the original paper by Schmal is in German, a language which I don't understand. Can someone please explain the proof of the theorem or at least point me to an open-source reference explaining the proof of the theorem in English?
